Question

If \(M_{\text {normal }}\) is the normal molecular mass and \(\alpha\) is the degree of ionization of \(\mathrm{K}_{3}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\), then the abnormal molecular mass of the complex in the solution will be: (a) \(M_{\text {normal }}(1+2 \alpha)^{-1}\) (b) \(M_{\text {normal }}(1+3 \alpha)^{-1}\) (c) \(M_{\text {normal }}(1+\alpha)^{-1}\) (d) equal to \(M_{\text {normal }}\)

Short answer

The abnormal molecular mass of the complex in solution will be \(M_{\text{normal}}(1 + 3 \alpha)^{-1}\), so the correct answer is (b) \(M_{\text{normal}}(1+3 \alpha)^{-1}\).

Step-by-step solution

Understand the Concept of Degree of Ionization

The degree of ionization, \(\alpha\), represents the fraction of a substance that dissociates into ions in solution. For a compound like \(\mathrm{K}_{3}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\), ionization would result in \(3\text{ K}^+\) ions and \(1 \left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{3-}\) ion.

Calculate the Total Number of Ions Formed

Upon dissociation, 1 molecule of \(\mathrm{K}_{3}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\) produces 4 ions in total (3 potassium ions and 1 complex ion).

Apply the Van't Hoff Factor

The Van't Hoff factor (i) gives an idea of the number of particles a compound forms in solution. If \(\alpha\) is the degree of ionization, the observed Van't Hoff factor for \(\mathrm{K}_{3}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\) is \(1 + 3\alpha\), because the compound dissociates into 4 parts, but since it is originally 1 part before dissociation, the change is 3 times \(\alpha\).

Calculate the Abnormal Molar Mass

The abnormal molar mass is calculated using the formula \(M_{\text{abnormal}} = \frac{M_{\text{normal}}}{i}\). Substitute \(i=1 + 3\alpha\) to obtain \(M_{\text{abnormal}} = \frac{M_{\text{normal}}}{1 + 3\alpha}\).

Key concepts

Degree of Ionization

Understanding the degree of ionization is essential when studying solutions and their properties. The degree of ionization, often represented by the Greek letter \(\alpha\), is the fraction of molecules that separate into ions when a substance is dissolved in a solvent. In simpler terms, it quantifies how much of a substance breaks down into its constituent ions in a solution.

For example, if you have a solution where just half of the molecules ionize, the degree of ionization \(\alpha\) would be 0.5. This concept becomes particularly notable when dealing with electrolytes, which can be classified as strong or weak based on their degree of ionization. Strong electrolytes completely dissociate into ions (\(\alpha\) is typically close to 1), while weak electrolytes do not fully dissociate (\(\alpha\) is much less than 1). Knowing the degree of ionization helps in predicting the behavior of the compound in solutions, including its conductivity and its reactivity.

Van't Hoff Factor

The Van't Hoff factor, symbolized by \(i\), plays a crucial role in determining the colligative properties of solutions. It is defined as the ratio of moles of particles in solution to the moles of solute dissolved. For non-electrolytes that do not dissociate in solution, the Van't Hoff factor is 1. However, for electrolytes which do dissociate, \(i\) can be greater than 1.

Take, for instance, NaCl: in water, it dissociates into two ions, Na+ and Cl-, so its Van't Hoff factor is approximately 2. However, in reality, not all NaCl molecules may dissociate. Accounting for this, the Van't Hoff factor can be represented as \(i = 1 + (n-1)\alpha\), where \(n\) is the number of ions formed when the molecule dissociates, and \(\alpha\) is the degree of ionization. Understanding the Van't Hoff factor helps predict how a solute will affect boiling point elevation, freezing point depression, and osmotic pressure of solutions.

Dissociation of Ionic Compounds

The process in which ionic compounds break apart into ions in a solvent is termed dissociation. This concept is of considerable importance for chemistry, as it influences how compounds interact in a solution. Ionic solids such as salts are composed of positive and negative ions held together by strong electrostatic forces. When these compounds are added to a solvent like water, these forces are weakened, causing the ions to separate and disperse throughout the solution.

Each ionic compound has a characteristic dissociation pattern. For example, when \(\mathrm{K}_3\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\) dissociates, it produces three \(\text{K}^+\) ions and one \(\left[\mathrm{Fu}(\mathrm{CN})_{6}\right]^{3-}\) ion. The ease with which a compound dissociates and the total number of ions produced influence its impact on the solution's properties, such as electrical conductivity and freezing point.

Molar Mass Calculation

The concept of molar mass calculation is fundamental in chemistry for converting between grams and moles of a substance. The molar mass is the weight of one mole (6.022 x \(10^{23}\)particles) of a substance and is expressed in grams per mole (g/mol). To calculate the molar mass of a compound, one adds the atomic masses of the individual elements in the compound, taking into account their respective numbers in the formula.

For instance, to calculate the molar mass of water (\(H_2O\)), you would use the atomic mass of hydrogen (approximately 1 g/mol) multiplied by 2, plus the atomic mass of oxygen (approximately 16 g/mol), which gives you about 18 g/mol for water. In cases where abnormal molecular masses are investigated, factors such as the dissociation into ions and the degree of ionization must be considered, which modify the straightforward molar mass calculation to account for these phenomena in solution.